(a) Block diagram of the system. Gene expression is calculated by means of a stochastic function, whilst fitness by means of a deterministic one. (b, c) Single-cell responses at different lactose doses (stochastic simulations, noise amplitudes of ηin = 0.5 and ηex = 0). Lactose and LacZ concentrations are denoted by x and y, respectively. The solid white line corresponds to the deterministic simulation. In b) the genotype contains a single copy of lacZ gene, whilst in c) it contains two copies. The value of mutual information (I) is shown in both cases: 1.29 bits of information in case of a singleton and 1.58 bits in case of a duplicate (about 25% increase in fidelity, significance assessed by a z-test, p ≈ 0 with 104 points). (d) Selection coefficient (S) of a genotype with two copies of lacZ gene over another with just one copy. The mean selection coefficient is shown (dashed line). Skewness coefficient of 2.63. W values calculated from x, y values shown in b, c). (e) Fitness (W) as a function of LacZ (constant x = 0.2 mM), showing the distributions of expression (boxplots) in case of one or two gene copies. The actual LacZ expression is shown (dashed line). (f) Dose-response curve between lactose concentration and the median LacZ expression (⟨y⟩). The solid lines correspond to the actual responses in case of one (black) or two (gray) gene copies (ηin = 0.5 and ηex = 0), whilst the dashed line corresponds to the optimal response. (g) Mean selection coefficient (⟨S⟩) landscape of gene duplication as a function of the median lactose dose (⟨x⟩, fluctuating dose) and the amplitude of intrinsic noise (ηin, with fixed ηex = 0.3). In all these plots, the expression levels of the duplicates with respect to the singletons are equal (ymax,1 = ymax,2 = 0.5).